Specific activity (aD)theor of a radionuclide D is defined as activity AD per unit mass MD of the radionuclide and is for a carrier-free radionuclide determined as follows (aD)theor = AD / MD = λDN / MD = λDNA / AD = (ln 2)NA / (t1/2)DAD , (12.134) where λD, (t1/2)D, and AD are the decay constant,

Question

Specific activity [latex]\left(a_D ight)_{theor}[/latex] of a radionuclide D is defined as activity [latex]\mathcal{A}_D[/latex] per unit mass [latex]M_D[/latex] of the radionuclide and is for a carrier-free radionuclide determined as follows

[latex]\left(a_{\mathrm{D}} ight)_{ ext {theor }}=\frac{\mathcal{A}_{\mathrm{D}}}{M_{\mathrm{D}}}=\frac{\lambda_{\mathrm{D}} N}{M_{\mathrm{D}}}=\frac{\lambda_{\mathrm{D}} N_{\mathrm{A}}}{A_{\mathrm{D}}}=\frac{(\ln 2) N_{\mathrm{A}}}{\left(t_{1 / 2} ight)_{\mathrm{D}} A_{\mathrm{D}}} ext {, }[/latex] (12.134)

where [latex]\lambda_{\mathrm{D}},\left(t_{1 / 2} ight)_{\mathrm{D}} ext {, and } A_{\mathrm{D}}[/latex] are the decay constant, half-life, and atomic mass, respectively, of the radionuclide D and [latex]N_A[/latex] is the Avogadro constant [latex]\left(6.022 imes 10^{23} \mathrm{~mol}^{-1} ight)[/latex].

(a) Derive expressions for specific activity [latex]a_D[/latex] of radionuclides produced in nuclear activation using: (1) saturation model of nuclear activation and (2) depletion model of nuclear activation. Discuss the dependence of [latex]a_D[/latex] on activation time t and particle fluence rate [latex]\dot{φ}[/latex]. What is the maximum specific activity [latex]\left(a_D ight)_{max}[/latex] achievable in nuclear activation using the saturation model and the depletion model?

(b) Calculate the theoretical specific activity [latex]\left(a_D ight)_{theor}[/latex] given in (12.134) for the following radionuclides D of interest in medical physics: cobalt-60 (Co-60), molybdenum-99 (Mo-99), europium-152 (Eu-152), iridium192 (Ir-192), and gold-198 (Au-198).

(c) Determine the maximum specific activity [latex]\left(a_D ight)_{max}[/latex] attainable in thermal neutron activation in a nuclear reactor with neutron fluence rate [latex]\dot{φ}[/latex] of [latex]5 imes 10^{11} \mathrm{~cm}^{-2} \cdot \mathrm{s}^{-1}, 2 imes 10^{13} \mathrm{~cm}^{-2} \cdot \mathrm{s}^{-1}, 3 imes 10^{14} \mathrm{~cm}^{-2} \cdot \mathrm{s}^{-1}[/latex], and [latex]1.2 imes 10^{16} \mathrm{~cm}^{-2} \cdot \mathrm{s}^{-1}[/latex] for the daughter nuclei of (b). Use both the saturation model and the depletion model of nuclear activation and compare the results obtained with the two models.

(d) Determine activation times [latex]\left(t_{max} ight)_D[/latex] required to attain maximum specific activities [latex]\left(a_D ight)_{max}[/latex] of daughter D radionuclides of (b) and neutron fluence rates [latex]\dot{φ}[/latex] of (c).

Accepted Answer

(a) The equation used for expressing nuclear activation with particle beams (T12.57) accounts for activation and depletion of parent P nuclei as well as for activation of daughter D nuclei. Activation of parent P and daughter D nuclei is governed by cross sections [latex]σ_P\ and\ σ_D[/latex], respectively; depletion of parent P nuclei is described by the difference between the number of parent P nuclei [latex]N_P(t)[/latex] at time t and the initial number [latex]N_P(0)[/latex] of parent nuclei P at time t = 0, where t stands for activation time.
Two approximations to the general activation equation [see (T12.57)] are in use:

(1) The activation process affects neither the daughter nuclei [latex]\left(σ_D = 0 ight)[/latex] nor the number [latex]N_P(t)[/latex] of parent nuclei [latex]\left[N_P(t) = N_P(0) = constant ight][/latex] resulting in the so-called saturation model of nuclear activation. The initial number [latex]N_P(0)[/latex] of parent nuclei is assumed infinitely large and not affected by nuclear activation of parent P into daughter D.
(2) The activation process does not affect the daughter nuclei ([latex]σ_D[/latex] = 0); however, it affects the number [latex]N_{\mathrm{P}}(t)[/latex] of parent nuclei [latex]\left[N_{\mathrm{P}}(0)>N_{\mathrm{P}}(t) eq ext { constant } ight][/latex] resulting in the so-called depletion model of nuclear activation. [latex]N_P(0)[/latex] is finite and undergoes depletion during the nuclear activation process.

(1) Saturation model: The general equation for normalized daughter activity [latex]y_{\mathrm{D}}(x) [/latex] as a function of normalized activation time [latex] x ext {, with } x=m \lambda_{\mathrm{D}} t / \ln 2=\sigma_{\mathrm{P}} \dot{\varphi} t / \ln 2[/latex], is for the saturation model of nuclear activation expressed as [see (T12.35)]

[latex]y_{\mathrm{D}}(x)=\frac{\mathcal{A}_{\mathrm{D}}(t)}{\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)}=\frac{\mathcal{A}_{\mathrm{D}}(t)}{\left(\mathcal{A}_{\mathrm{D}} ight)_{\mathrm{sat}}}=1-e^{-\frac{x}{m} \ln 2},[/latex] (12.135)

where

[latex]\left(\mathcal{A}_{\mathrm{D}} ight)_{ ext {sat }}[/latex] is the saturation activity of the daughter defined as [latex]\left(\mathcal{A}_{\mathrm{D}} ight)_{ ext {sat }}=\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)[/latex].

[latex]\dot{\varphi}[/latex] is the particle fluence rate.

m is the activation factor defined as [latex]m=\sigma_{\mathrm{P}} \dot{\varphi} / \lambda_{\mathrm{D}}[/latex].

After insertion of (12.135) into (12.134) the specific activity [latex]a_D[/latex] for the saturation model is given by the following expression

[latex]\begin{aligned} a_{\mathrm{D}} & =\frac{\mathcal{A}_{\mathrm{D}}(t)}{M_{\mathrm{P}}}=\frac{\left(\mathcal{A}_{\mathrm{D}} ight)_{\mathrm{sat}} y_{\mathrm{D}}(x)}{M_{\mathrm{P}}}=\frac{\left(\mathcal{A}_{\mathrm{D}} ight)_{\mathrm{sat}}}{M_{\mathrm{P}}}\left\{1-e^{-\frac{x}{m} \ln 2} ight\}=\frac{\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)}{M_{\mathrm{P}}}\left\{1-e^{-\frac{x}{m} \ln 2} ight\} \ & =\frac{\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{A}}}{A_{\mathrm{P}}}\left\{1-e^{-\frac{x}{m} \ln 2} ight\}=\frac{\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{A}}}{A_{\mathrm{P}}}\left\{1-\frac{1}{2^{x / m}} ight\},\quad (12.136) \end{aligned}[/latex]

where we used the standard expression for [latex]N_P(0)/M_P = N_A/A_P\ with\ M_P[/latex] the initial mass of the parent P nuclide and [latex]A_P[/latex] the atomic mass of the parent nuclide.
From (12.136) we note that the maximum activity [latex]\left(a_D ight)_{max}[/latex] for x → ∞ is proportional to the particle fluence rate [latex]\dot{\varphi}[/latex] and the proportionality constant is [latex]\left(\sigma_{\mathrm{P}} N_{\mathrm{A}} / A_{\mathrm{P}} ight)[/latex].

[latex]\left(a_{\mathrm{D}} ight)_{\max }=\frac{\sigma_{\mathrm{P}} N_{\mathrm{A}}}{A_{\mathrm{P}}} \dot{\varphi} .[/latex] (12.137)

The linear relationship between [latex]\left(a_{\mathrm{D}} ight)_{\max } ext { and } \dot{\varphi} ext { works fine at relatively low } \dot{\varphi} ext {; }[/latex] however, at [latex]\dot{\varphi} ightarrow \infty,(12.137) ext { suggests that }\left(a_{\mathrm{D}} ight)_{\max } ightarrow \infty[/latex] in contradiction with the theoretical specific activity [latex]\left(a_D ight)_{theor}[/latex] of (12.134) that is well defined, independent of [latex]\dot{φ}[/latex], and finite for a given radionuclide.

(2) Depletion model: The general equation for normalized daughter activity [latex]y_{\mathrm{D}}(x)[/latex] as a function of normalized activation time [latex] x ext {, with } x=m \lambda_{\mathrm{D}} t / \ln 2=\sigma_{\mathrm{P}} \dot{\varphi} t / \ln 2[/latex], is for the depletion model of nuclear activation expressed as [see (T12.33)]

[latex]y_{\mathrm{D}}(x)=\frac{\mathcal{A}_{\mathrm{D}}(t)}{\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0)}=\frac{1}{1-m}\left\{e^{-x \ln 2}-e^{-\frac{x}{m} \ln 2} ight\}=\frac{1}{1-m}\left\{\frac{1}{2^x}-\frac{1}{2^{x / m}} ight\},[/latex] (12.138)

where m is the activation factor defined as [latex]m=\sigma_{\mathrm{P}} \dot{\varphi} / \lambda_{\mathrm{D}} .[/latex]
After insertion of (12.138) into (12.134) the specific activity [latex]a_D[/latex] for the depletion model is given by the following

[latex]a_{\mathrm{D}}=\frac{\mathcal{A}_{\mathrm{D}}(t)}{M_{\mathrm{P}}}=\frac{\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{P}}(0) y_{\mathrm{D}}(x)}{M_{\mathrm{P}}}=\frac{\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{A}} y_{\mathrm{D}}(x)}{A_{\mathrm{P}}}=\frac{\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{A}}}{A_{\mathrm{P}}(1-m)}\left\{e^{-x \ln 2}-e^{-\frac{x}{m} \ln 2} ight\},[/latex] (12.139)

where, as in (12.136), we used the relationship [latex]N_{\mathrm{P}}(0) / M_{\mathrm{P}}=N_{\mathrm{A}} / A_{\mathrm{P}} .[/latex]
In contrast to (12.136) which exhibits [latex]\left(a_D ight)_{max}[/latex] given by (12.137) at x → ∞, (12.139) exhibits maximum specific activity [latex]\left(a_D ight)_{max}[/latex] at the point of ideal equilibrium between [latex]y_P\ and\ y_D[/latex] which occurs at [latex]\left[\left(x_D ight)_{max},\left(y_D ight)_{max} ight][/latex], as shown in (T12.29 through (T12.32)), while [latex]\lim _{x ightarrow \infty}\left(a_{\mathrm{D}} ight)_{\max }=0[/latex].
The maximum specific activity [latex]\left(a_D ight)_{max}[/latex] calculated for the depletion model is from (12.139) now written as follows

[latex]\left(a_{\mathrm{D}} ight)_{\max }=\frac{\sigma_{\mathrm{P}} \dot{\varphi} N_{\mathrm{A}}\left(y_{\mathrm{D}} ight)_{\max }}{A_{\mathrm{P}}}=\frac{\sigma_{\mathrm{P}} N_{\mathrm{A}}}{A_{\mathrm{P}}} \dot{\varphi} e^{\frac{m}{1-m} \ln m}=\frac{\sigma_{\mathrm{P}} N_{\mathrm{A}}}{A_{\mathrm{P}}} \dot{\varphi} \exp \left\{-\frac{\frac{\sigma_{\mathrm{P}} \dot{\varphi}}{\lambda_{\mathrm{D}}}}{\frac{\sigma_{\mathrm{P}} \dot{\varphi}}{\lambda_{\mathrm{D}}}-1} \ln \frac{\sigma_{\mathrm{P}} \dot{\varphi}}{\lambda_{\mathrm{D}}} ight\}[/latex] (12.140)

where we used the definition of [latex]m=\sigma_{\mathrm{P}} \dot{\varphi} / \lambda_{\mathrm{D}} ext { and the expression for }\left(y_{\mathrm{D}} ight)_{\max }[/latex] derived in (12.131) of Prob. 255.

(b) Specific activity [latex]a_D[/latex] of daughter radionuclide D produced in nuclear activation of parent nuclide P depends on various activation parameters, such as: activation cross section [latex]σ_P[/latex] of the parent P nucleus, decay constant [latex]λ_D[/latex] of the daughter D nucleus, activation particle fluence rate [latex]\dot{φ}[/latex], and activation time t. For a given parent– daughter nuclear configuration, [latex]σ_P\ and\ λ_D[/latex] are fixed, so that the specific activity [latex]a_D[/latex] depends on [latex]\dot{φ}[/latex] and t, as shown for the saturation model in (12.136) and for the depletion model in (12.139). However, as shown in Prob. 258, the maximum attainable specific activity [latex]\left(a_D ight)_{max}[/latex] is limited by the theoretical specific activity [latex]\left(a_D ight)_{theor}[/latex] given in (12.134) which defines the upper limit of achievable specific activities in nuclear activation for a given parent–daughter configuration.
Theoretical specific activities [latex]\left(a_D ight)_{theor}[/latex] of various radionuclides of importance in medical physics (Co-60, Mo-99, Eu-152, Ir-192, and Au-198) are calculated with (12.134) as follows (thermal neutron cross sections [latex]σ_D[/latex] are from the IAEA TRS Report #156)

(1) Cobalt-60: [latex]\left(t_{1 / 2} ight)_{\mathrm{Co}-60}=5.26 \mathrm{a}, A_{\mathrm{Co}-60}=59.93 \mathrm{~g} / \mathrm{mol}[/latex]
Parent nucleus: cobalt-59, [latex]\sigma_{\mathrm{Co}-59}=37.2 \mathrm{~b}[/latex].

[latex]\begin{aligned} \left(a_{ ext {Co-60 }} ight)_{ ext {theor }} & =\frac{(\ln 2) N_{\mathrm{A}}}{\left(t_{1 / 2} ight)_{\mathrm{Co}-60} A_{\mathrm{Co}-60}} \ & =\frac{(\ln 2) imes\left(6.022 imes 10^{23} \mathrm{~mol}^{-1} ight)}{(5.26 \mathrm{a}) imes(365 \mathrm{~d} / \mathrm{a}) imes(24 \mathrm{~h} / \mathrm{d}) imes(3600 \mathrm{~s} / \mathrm{h}) imes\left(59.93 \mathrm{~g} \cdot \mathrm{mol}^{-1} ight)} \ & =4.199 imes 10^{13} \mathrm{~Bq} / \mathrm{g}=42.99 \mathrm{TBq} / \mathrm{g}=\frac{4.199 imes 10^{13} \mathrm{~Bq} / \mathrm{g}}{3.7 imes 10^{10} \mathrm{~Bq} / \mathrm{Ci}} \ & =1.135 imes 10^3 \mathrm{Ci} / \mathrm{g} .\quad (12.141) \end{aligned}[/latex]

(2) [latex] ext { Molybdenum-99: }\left(t_{1 / 2} ight)_{\mathrm{Mo}-99}=65.94 \mathrm{~h}, A_{\mathrm{Mo}-99}=98.91 \mathrm{~g} / \mathrm{mol}[/latex]
Parent nucleus: molybdenum-98, [latex]\sigma_{\mathrm{Mo}-98}=0.13[/latex] b

[latex]\begin{aligned} \left(a_{ ext {Mo-99 }} ight)_{ ext {theor }} & =\frac{(\ln 2) N_{\mathrm{A}}}{\left(t_{1 / 2} ight)_{\mathrm{Mo}-99} A_{\mathrm{Mo}-99}} \ & =\frac{(\ln 2) imes\left(6.022 imes 10^{23} \mathrm{~mol}^{-1} ight)}{(65.94 \mathrm{~h}) imes(3600 \mathrm{~s} / \mathrm{h}) imes\left(98.91 \mathrm{~g} \cdot \mathrm{mol}^{-1} ight)} \ & =1.778 imes 10^{16} \mathrm{~Bq} / \mathrm{g}=\frac{1.778 imes 10^{16} \mathrm{~Bq} / \mathrm{g}}{3.7 imes 10^{10} \mathrm{~Bq} / \mathrm{Ci}} \ & =4.805 imes 10^5 \mathrm{Ci} / \mathrm{g} .\quad (12.142) \end{aligned}[/latex]

(3) [latex] ext { Europium-152: }\left(t_{1 / 2} ight)_{\mathrm{Eu}-152}=13.54 \mathrm{a}, A_{\mathrm{Eu}-152}=151.92 \mathrm{~g} / \mathrm{mol}[/latex]
Parent nucleus: europium-151, [latex]\sigma_{\mathrm{Eu}-151}=5300[/latex] b

[latex]\begin{aligned} \left(a_{\mathrm{Eu}-152} ight)_{ ext {theor }} & =\frac{(\ln 2) N_{\mathrm{A}}}{\left(t_{1 / 2} ight)_{\mathrm{Eu}-152} A_{\mathrm{Eu}-152}} \ & =\frac{(\ln 2) imes\left(6.022 imes 10^{23} \mathrm{~mol}^{-1} ight)}{(13.54 \mathrm{a}) imes(365 \mathrm{~d} / \mathrm{a}) imes(24 \mathrm{~h} / \mathrm{d}) imes(3600 \mathrm{~s} / \mathrm{h}) imes\left(151.92 \mathrm{~g} \cdot \mathrm{mol}^{-1} ight)} \ & =6.434 imes 10^{12} \mathrm{~Bq} / \mathrm{g}=6.434 \mathrm{TBq} / \mathrm{g}=\frac{6.434 imes 10^{12} \mathrm{~Bq} / \mathrm{g}}{3.7 imes 10^{10} \mathrm{~Bq} / \mathrm{Ci}} \ & =1.739 imes 10^2 \mathrm{Ci} / \mathrm{g} .\quad (12.143) \end{aligned}[/latex]

(4) [latex] ext { Iridium-192: }\left(t_{1 / 2} ight)_{\mathrm{Ir}-192}=73.8 \mathrm{~d}, A_{\mathrm{Mo}-99}=98.91 \mathrm{~g} / \mathrm{mol}[/latex]
Parent nucleus: iridium-191, [latex]\sigma_{\mathrm{Ir}-191}=954[/latex] b

[latex]\begin{aligned} \left(a_{\mathrm{Ir}-192} ight)_{ ext {theor }} & =\frac{(\ln 2) N_{\mathrm{A}}}{\left(t_{1 / 2} ight)_{\mathrm{Ir}-192} A_{\mathrm{Ir}-192}} \ & =\frac{(\ln 2) imes\left(6.022 imes 10^{23} \mathrm{~mol}^{-1} ight)}{(73.8 \mathrm{~d}) imes(24 \mathrm{~h} / \mathrm{d}) imes(3600 \mathrm{~s} / \mathrm{h}) imes\left(191.96 \mathrm{~g} \cdot \mathrm{mol}^{-1} ight)} \ & =3.410 imes 10^{14} \mathrm{~Bq} / \mathrm{g}=341 \mathrm{TBq} / \mathrm{g}=\frac{3.410 imes 10^{12} \mathrm{~Bq} / \mathrm{g}}{3.7 imes 10^{10} \mathrm{~Bq} / \mathrm{Ci}} \ & =9.217 imes 10^3 \mathrm{Ci} / \mathrm{g} .\quad (12.144) \end{aligned}[/latex]

(5) [latex] ext { Gold-198: }\left(t_{1 / 2} ight)_{\mathrm{Au}-198}=64.68 \mathrm{~h}, A_{\mathrm{Au}-198}=197.97 \mathrm{~g} / \mathrm{mol}[/latex]
Parent nucleus: gold-197, [latex]\sigma_{\mathrm{Au}-197}=98.8[/latex] b

[latex]\begin{aligned} \left(a_{\mathrm{Au}-198} ight)_{ ext {theor }} & =\frac{(\ln 2) N_{\mathrm{A}}}{\left(t_{1 / 2} ight)_{\mathrm{Au}-198} A_{\mathrm{Au}-198}} \ & =\frac{(\ln 2) imes\left(6.022 imes 10^{23} \mathrm{~mol}^{-1} ight)}{(64.68 \mathrm{~h}) imes(3600 \mathrm{~s} / \mathrm{h}) imes\left(197.97 \mathrm{~g} \cdot \mathrm{mol}^{-1} ight)} \end{aligned}\ \begin{aligned} & =9.055 imes 10^{15} \mathrm{~Bq} / \mathrm{g}=\frac{9.055 imes 10^{15} \mathrm{~Bq} / \mathrm{g}}{3.7 imes 10^{10} \mathrm{~Bq} / \mathrm{Ci}} \ & =2.447 imes 10^5 \mathrm{Ci} / \mathrm{g} .\quad (12.145) \end{aligned}[/latex]

(c) Maximum specific activities [latex]\left(a_D ight)_{max}[/latex] of radionuclide D attainable in nuclear activation were derived in (a) and are given in (12.137) and (12.140) for the saturation model and the depletion model, respectively. We will now use these two expressions to determine [latex]\left(a_D ight)_{max}[/latex] of (1) Co-60, (2) Mo-99, (3) Eu-152, (4) Ir-192, and (5) Au-198 for four neutron fluence rates: [latex]5 imes 10^{11} \mathrm{~cm}^{-2} \cdot \mathrm{s}^{-1}, 2 imes 10^{13} \mathrm{~cm}^{-2} \cdot \mathrm{s}^{-1}, 3 imes 10^{14} \mathrm{~cm}^{-2} \cdot \mathrm{s}^{-1} ext {, and } 1.2 imes 10^{16} \mathrm{~cm}^{-2} \cdot \mathrm{s}^{-1}[/latex]. Before embarking on calculation of [latex]\left(a_D ight)_{max}[/latex], we summarize in Table 12.5 the relevant physical data, obtained from Appendix A or from the literature for the five radionuclides. Entries for [latex]\left(a_D ight)_{theor}[/latex] were determined in (b) with (12.134). In Table 12.6 we list activation factors m for the five radionuclides and four thermal neutron fluence [latex]\dot{φ}[/latex] rates. It is evident that the range in [latex]\dot{φ}[/latex] in nature is quite large in our example extending from about [latex]∼2 imes 10^{−8}\ to\ ∼4 imes 10^4[/latex]. Entries for [latex]\left(a_D ight)_{theor}[/latex] were calculated in (b) with (12.134).
Results of [latex]\left(a_D ight)_{max}[/latex] calculations are summarized for radionuclides cobalt-60, molybdenum-99, europium-152, iridium-192, and gold-198 in Table 12.7 for the saturation model determined with (12.137) and in Table 12.8 for the depletion model determined with (12.140). For a given radionuclide the maximum specific activity [latex]\left(a_D ight)_{max}[/latex] in Table 12.7 is linearly proportional with [latex]\dot{φ}[/latex] in the whole fluence rate range, while in Table 12.8 the maximum specific activity [latex]\left(a_D ight)_{max}[/latex] is proportional to [latex]\dot{φ}[/latex] at low [latex]\dot{φ}[/latex] and saturates at [latex]\left(a_D ight)_{theor}[/latex] at high [latex]\dot{φ}[/latex]. Results displayed in Tables 12.7 and 12.8 are plotted in Fig. 12.4, for the saturation model with light solid lines and for the depletion model with heavy solid curves.
The practical range of thermal neutron fluence [latex]\dot{φ}[/latex] in nuclear reactors ranges from [latex]\sim 10^{11} \mathrm{~cm}^{-2} \cdot \mathrm{s}^{-1} ext { to } \sim 10^{16} \mathrm{~cm}^{-2} \cdot \mathrm{s}^{-1}[/latex]. The range covered in Fig. 12.4 is an expanded practical range to illustrate the behavior of [latex]\left(a_D ight)_{max}[/latex] that would be observed with saturation and depletion models if the practical fluence rate range were expanded to [latex]\sim 10^{22} \mathrm{~cm}^{-2} \cdot \mathrm{s}^{-1}[/latex]. It is obvious that in the practical range of [latex]\dot{φ}[/latex] activation of Mo-99 and to a large extent also of Au-198 can be adequately described by the saturation model of nuclear activation. Ir-192 and Co-60 follow the saturation model for [latex]\dot{φ} < ∼ 10^{13}\ cm^{−2}\ ·\ s^{−1}[/latex] but above [latex]10^{13}\ cm^{−2}\ ·\ s^{−1}[/latex] should be described with the depletion model. Eu-152 for all practical fluence rates [latex]\dot{φ}[/latex] should be described with the depletion model.

(d) Activation time [latex]\left(t_{max} ight)_D[/latex] required to attain the maximum specific activity [latex]\left(a_D ight)_{max}[/latex] of daughter D radionuclide at a given neutron fluence rate [latex]\dot{φ}[/latex] is determined from the expression for [latex]\left(x_D ight)_{max}[/latex] which gives the normalized time coordinate of the point of ideal parent-daughter equilibrium, derived in (12.88) of Prob. 253 and in (T12.29). The following relationship holds for [latex]\left(x_D ight)_{max}\ and\ \left(t_{max} ight)_D[/latex].

[latex]\left(x_{\mathrm{D}} ight)_{\max }=\frac{m}{m-1} \frac{\ln m}{\ln 2}=\frac{m \lambda_{\mathrm{D}}\left(t_{\max } ight)_{\mathrm{D}}}{\ln 2} \quad ext { or } \quad\left(t_{\max } ight)_{\mathrm{D}}=\frac{\left(x_{\mathrm{D}} ight)_{\max } \ln 2}{m \lambda_{\mathrm{D}}}=\frac{\ln m}{(m-1) \lambda_{\mathrm{D}}} ext {, }[/latex] (12.146)

where m is the activation factor defined as [latex]m=\sigma_{\mathrm{P}} \dot{\varphi} / \lambda_{\mathrm{D}}, \sigma_{\mathrm{P}}[/latex] is the activation cross section of the parent nuclide, and [latex]λ_D[/latex] is the decay constant of the daughter radionuclide. Results of [latex]\left(t_{max} ight)_D[/latex] calculation using (12.146) are summarized in Table 12.9 and show that, for a given daughter radionuclide D, ([latex]t_{max})_D[/latex] is roughly inversely proportional to the particle fluence rate [latex]\dot{φ}[/latex]. Thus, the higher is the thermal neutron fluence [latex]\dot{φ}[/latex], the shorter is the activation time [latex]\left(t_{max} ight)_D[/latex] required to attain the maximum specific activity [latex]\left(a_D ight)_{max}[/latex] for a given daughter D radionuclide.

Question 12.6.Q6: Specific activity (aD)theor of a radionuclide D is defined a......

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